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Putnam
1967 Putnam
A1
Putnam 1967 A1
Putnam 1967 A1
Source: Putnam 1967
May 13, 2022
Putnam
trigonometry
inequalities
Problem Statement
Let
f
(
x
)
=
a
1
sin
x
+
a
2
sin
2
x
+
⋯
+
a
n
sin
n
x
f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx
f
(
x
)
=
a
1
sin
x
+
a
2
sin
2
x
+
⋯
+
a
n
sin
n
x
, where
a
1
,
a
2
,
…
,
a
n
a_1 ,a_2 ,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
are real numbers and where
n
n
n
is a positive integer. Given that
∣
f
(
x
)
∣
≤
∣
sin
x
∣
|f(x)| \leq | \sin x |
∣
f
(
x
)
∣
≤
∣
sin
x
∣
for all real
x
,
x,
x
,
prove that
∣
a
1
+
2
a
2
+
⋯
+
n
a
n
∣
≤
1.
|a_1 +2a_2 +\cdots +na_{n}|\leq 1.
∣
a
1
+
2
a
2
+
⋯
+
n
a
n
∣
≤
1.
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